Related papers: Extending Upper Cluster Algebras
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
We consider an orbit category of the bounded derived category of a path algebra of type A_n which can be viewed as a -(m+1)-cluster category, for m >= 1. In particular, we give a characterisation of those maximal m-rigid objects whose…
We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group…
Recently, Ramos and Whiting showed that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of a certain cluster algebra. Based on their idea and method, we show that the same property holds for any…
We develop a new approach to highest weight categories $\cal{C}$ with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For $\cal{C}$ admitting…
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of…
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander--Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with…
Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is…
In the construction of a cluster algebra on the homogeneous coordinate ring of a partial flag variety by Gei{\ss}, Leclerc and Schr{\"{o}}er, they defined a special map denoted by ``tilde". This map lifts each element $f$ of the coordinate…
We explore the question concerning the number of distinct resonant algebras depending on the generator content, which consists of the Lorentz generator, translation, and new additional Lorentz-like and translation-like generators. Such…
We study the cluster algebras arising from cluster tubes with rank bigger than $1$. Cluster tubes are $2-$Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
We first introduce and study the notion of semi-regular flat modules, and then show that a ring $R$ is a strong \Prufer\ ring if and only if every submodule of a semi-regular flat $R$-module is semi-regular flat, if and only if every ideal…
A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar 2--co nnected finite or countable graph $G$ of girth $r$ and maximal vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the plane, denote it by…
We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\zS$ in a…
For any ring $R$, we introduce an invariant in the form of a partially ordered abelian semigroup $\mathrm{S}(R)$ built from an equivalence relation on the class of countably generated projective modules. We call $\mathrm{S}(R)$ the Cuntz…
Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting…
The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the…
We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a notion of quiver mutation that is compatible with quasi-cluster algebra mutation defined by Dupont and Palesi. Moreover, we use our quiver to…